quadratic form
Noun
Meaning
(mathematics, number theory, algebra) A homogeneous polynomial of degree 2 in a given number of variables.
(statistics, multivariate statistics) A scalar quantity of the form
ε
T
Λ
ε
{\displaystyle \varepsilon ^{T}\Lambda \varepsilon }
, where
ε
{\displaystyle \varepsilon }
is a vector of n random variables, and
Λ
{\displaystyle \Lambda }
is an n-dimensional symmetric matrix.
Examples
Although there are several more intrinsic constructions, the spin representations are not functorial in the quadratic form, so they cannot be built up naturally within the tensor algebra.
Conic section and quadratic form Diagram, description, and definitions Cone with cross-sections (To enlarge, click on diagram.
However, the discriminant in a PDE is given by due to the convention of the xy term being 2B rather than B; formally, the discriminant (of the associated quadratic form) is with the factor of 4 dropped for simplicity.
Component spinors Given a vector space V and a quadratic form g an explicit matrix representation of the Clifford algebra Cℓ(V, g) can be defined as follows.
Further properties Expectation of a quadratic form One can take the expectation of a quadratic form in the random vector X as follows: Kendrick, David, Stochastic Control for Economic Models, McGraw-Hill, 1981.
Indeed, the linear map on V defined by v ↦ −v ( reflection through the origin ) preserves the quadratic form Q and so by the universal property of Clifford algebras extends to an algebra automorphism :α: Cℓ(V, Q) → Cℓ(V, Q).